We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators wi... more We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ [ - 1 , 1 ] and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ ( 1 - | x | β ) - 1 / 2 when the non-perturbed potential resembles $$|x|^\beta $$ | x | β , $$\beta >0$$ β > 0 , for large x, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.
We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal boun... more We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half-plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large-parameter regime.
Recently the authors constructed [1, an example of a nuclear F-space X without a base. This resol... more Recently the authors constructed [1, an example of a nuclear F-space X without a base. This resolved one of the old questions in the theory of nuclear spaces (and inthe geometric theory of linear operators). We remark that in contrast to the ]3anach case, where Enflo's counterexamples (also see surveys [7, 8]) deny the approximation property and, by the same token, automatically, also the presence of a base, the space X, as every nuclear F-space (for example, see [9], Statement III.9.2), has the approximation property and the absence of a base in X is established in with the vital use of the commutativity of the Boolean algebra of projectors. The aim of the present paper is the construction of a continual family of pairwise nonisomorphic nuclear F-spaces without base. In the constructions and in the proofs we shall use not only the authors' results in but also the constructions therein, sometimes even in great detail; therefore, in the remaining paragraphs of the present article it is assumed that the reader is acquainted with these constructions. This refers to 82182 5-9 where the absence of the base is proved, but the construction of the continuum of pairwise nonisomorphic spaces (in 82182 2-4) is based on other (not so analytic) considerations developed earlier by Dragilev [10], Mityagin [4, and Zakharyuta [12].
... is equivalent to the original system (see [2] or §2 in [13]). If X is nuclear, then, as Dynin... more ... is equivalent to the original system (see [2] or §2 in [13]). If X is nuclear, then, as Dynin and Mityagin showed in [7] (for the proof see §4 in [12] or §10.2 in [18]), each basis in X is unconditional (and absolute). ... 0(E, F; A, B) ---- max S sup inf Ux--YI[B; [ a~E yCzF nx[iA < I ...
For one-dimensional Dirac operators subject to periodic or antiperiodic boundary conditions, we g... more For one-dimensional Dirac operators subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in L 2 ([0, π], C 2 ). In particular, if the potential matrix v is skew-symmetric (i.e., Q = -P ), or more generally if Q = tP for some real t = 0, then there exists a Riesz basis that consists of root functions of the operator L.
We consider the limit measures induced by the rescaled eigenfunctions of certain perturbed single... more We consider the limit measures induced by the rescaled eigenfunctions of certain perturbed single-well Schrodinger operators. We show that the limit measure is proportional to $(1-|x|^\beta)_+^{-1/2} \mathrm{d} x$ when the non-perturbed potential resembles $|x|^\beta$ for large $x$ and $\beta \in [1, \infty)$ and to $\mathbbm{1}_{[-1,1]}(x) \mathrm{d} x$ for faster growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.
PLAMEN DJAKOV AND BORIS MITYAGIN 4. Matrix representation of L bc and its resolvent R bc (λ) 5. L... more PLAMEN DJAKOV AND BORIS MITYAGIN 4. Matrix representation of L bc and its resolvent R bc (λ) 5. Localization of spectra 6. Bari-Markus property in the case of strictly regular boundary conditions 7. Bari-Markus property in the case of regular but not strictly regular boundary conditions 8. Miscellaneous; pointwise convergence and equiconvergence
Let γ n denote the length of the nth zone of instability of the Hill operator Ly = −y − [4tα cos ... more Let γ n denote the length of the nth zone of instability of the Hill operator Ly = −y − [4tα cos 2x + 2α 2 cos 4x]y, where α = 0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α → 0 γ n = 8α n 2 n [(n − 1)!] 2
Consider the Schrödinger equation −y″+V(x)y=λy with a periodic real-valued L2-potential V of peri... more Consider the Schrödinger equation −y″+V(x)y=λy with a periodic real-valued L2-potential V of period 1, V(x)=∑m=−∞∞v(m) exp(2πimx). Let {λn−,λn+} be its zones of instability, i.e. {λn−,λn+} are pairs of periodic and antiperiodic eigenvalues, and b∈(0,1) determines Gevrey classes.THEOREM.If∑n=1∞∣λn+−λn−∣2(1+n)2se2anb&amp;amp;amp;amp;amp;amp;lt;∞s⩾0, a0 , then V(x) is a Gevrey function, and moreover∑m=−∞∞∣v(m)∣2(1+∣m∣)2se2a∣m∣b&amp;amp;amp;amp;amp;amp;lt;∞.
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular pote... more We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L 2-potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p-spaces and other rearrangement invariant function spaces.
We consider the Hill operator Ly = −y ′′ + v(x)y, 0 ≤ x ≤ π, subject to periodic or antiperiodic ... more We consider the Hill operator Ly = −y ′′ + v(x)y, 0 ≤ x ≤ π, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the form v(x) = ae −2irx + be 2isx , a, b = 0, r, s ∈ N, r = s. It is shown that the system of root functions does not contain a basis in L 2 ([0, π], C) if bc are periodic or if bc are antiperiodic and r, s are odd or r = 1 and s ≥ 3.
Let us consider the Dirac operatorL=iJddx+U,J=100-1,U=0acos2πxacos2πx0,where a≠0 is real, on I=[0... more Let us consider the Dirac operatorL=iJddx+U,J=100-1,U=0acos2πxacos2πx0,where a≠0 is real, on I=[0,1] with boundary conditions bc=Per+, i.e., F(1)=F(0), and bc=Per-, i.e., F(1)=-F(0), F=f1f2∈H1(I). Then σ(Lbc)=-σ(Lbc), and all λ∈σPer+(L(U)) are of multiplicity 2, while λ∈σPer-(L(U)) are simple (Theorem 15). This is an analogue of Ince&amp;#39;s statement for Mathieu–Hill operator.Links between the spectra of Dirac and Hill operators lead to detailed information about
We give a complete description of the structure of the spectra of Hill operator Ly = −y + (a cos ... more We give a complete description of the structure of the spectra of Hill operator Ly = −y + (a cos 2x + b cos 4x)y, a, b real, x ∈ [0, ] with periodic or antiperiodic boundary conditions. As in Ince [Proc. London Math. Soc. 23 (1923) 56] and Magnus-Winkler [Hill's Equation, Interscience Publishers, Wiley, 1969], properties and spectra of special tridiagonal matrices is a core of our analysis.
It is proved using positive definite functions that a normed space X is unifomly homeomorphic to ... more It is proved using positive definite functions that a normed space X is unifomly homeomorphic to a subset of a Hilbert space, if and only if X is (linearly) isomorphic to a subspace of a L,,(/z) space (= the space of the measurable functions on a probability space with convergence in probability). As a result we get that lp (respectively L,(0,1)), 2<p <~, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces.
A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] l... more A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] led to an integral operator K-see (1), (2). They guessed that the equation E 0 (a, T) = 1 where E 0 is the largest eigenvalue of the operator K has a solution (*) T (a) = 1 − τ (a) with τ (a) ∼ a 2/5 when a goes to 0. τ (a) imitates the shift of critical (instability) temperature. We give a rigorous analysis of an anisotropic integral operator K and prove the asymptotic (*)-see Theorem 8 and Proposition 10. Additive Uncertainty Principle (of Landau-Pollack-Slepian [SP], [LP1, LP2]) plays important role in this analysis.
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular pote... more We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L 2-potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p-spaces and other rearrangement invariant function spaces.
We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators wi... more We consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ [ - 1 , 1 ] and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ ( 1 - | x | β ) - 1 / 2 when the non-perturbed potential resembles $$|x|^\beta $$ | x | β , $$\beta >0$$ β > 0 , for large x, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.
We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal boun... more We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half-plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large-parameter regime.
Recently the authors constructed [1, an example of a nuclear F-space X without a base. This resol... more Recently the authors constructed [1, an example of a nuclear F-space X without a base. This resolved one of the old questions in the theory of nuclear spaces (and inthe geometric theory of linear operators). We remark that in contrast to the ]3anach case, where Enflo's counterexamples (also see surveys [7, 8]) deny the approximation property and, by the same token, automatically, also the presence of a base, the space X, as every nuclear F-space (for example, see [9], Statement III.9.2), has the approximation property and the absence of a base in X is established in with the vital use of the commutativity of the Boolean algebra of projectors. The aim of the present paper is the construction of a continual family of pairwise nonisomorphic nuclear F-spaces without base. In the constructions and in the proofs we shall use not only the authors' results in but also the constructions therein, sometimes even in great detail; therefore, in the remaining paragraphs of the present article it is assumed that the reader is acquainted with these constructions. This refers to 82182 5-9 where the absence of the base is proved, but the construction of the continuum of pairwise nonisomorphic spaces (in 82182 2-4) is based on other (not so analytic) considerations developed earlier by Dragilev [10], Mityagin [4, and Zakharyuta [12].
... is equivalent to the original system (see [2] or §2 in [13]). If X is nuclear, then, as Dynin... more ... is equivalent to the original system (see [2] or §2 in [13]). If X is nuclear, then, as Dynin and Mityagin showed in [7] (for the proof see §4 in [12] or §10.2 in [18]), each basis in X is unconditional (and absolute). ... 0(E, F; A, B) ---- max S sup inf Ux--YI[B; [ a~E yCzF nx[iA < I ...
For one-dimensional Dirac operators subject to periodic or antiperiodic boundary conditions, we g... more For one-dimensional Dirac operators subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in L 2 ([0, π], C 2 ). In particular, if the potential matrix v is skew-symmetric (i.e., Q = -P ), or more generally if Q = tP for some real t = 0, then there exists a Riesz basis that consists of root functions of the operator L.
We consider the limit measures induced by the rescaled eigenfunctions of certain perturbed single... more We consider the limit measures induced by the rescaled eigenfunctions of certain perturbed single-well Schrodinger operators. We show that the limit measure is proportional to $(1-|x|^\beta)_+^{-1/2} \mathrm{d} x$ when the non-perturbed potential resembles $|x|^\beta$ for large $x$ and $\beta \in [1, \infty)$ and to $\mathbbm{1}_{[-1,1]}(x) \mathrm{d} x$ for faster growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.
PLAMEN DJAKOV AND BORIS MITYAGIN 4. Matrix representation of L bc and its resolvent R bc (λ) 5. L... more PLAMEN DJAKOV AND BORIS MITYAGIN 4. Matrix representation of L bc and its resolvent R bc (λ) 5. Localization of spectra 6. Bari-Markus property in the case of strictly regular boundary conditions 7. Bari-Markus property in the case of regular but not strictly regular boundary conditions 8. Miscellaneous; pointwise convergence and equiconvergence
Let γ n denote the length of the nth zone of instability of the Hill operator Ly = −y − [4tα cos ... more Let γ n denote the length of the nth zone of instability of the Hill operator Ly = −y − [4tα cos 2x + 2α 2 cos 4x]y, where α = 0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α → 0 γ n = 8α n 2 n [(n − 1)!] 2
Consider the Schrödinger equation −y″+V(x)y=λy with a periodic real-valued L2-potential V of peri... more Consider the Schrödinger equation −y″+V(x)y=λy with a periodic real-valued L2-potential V of period 1, V(x)=∑m=−∞∞v(m) exp(2πimx). Let {λn−,λn+} be its zones of instability, i.e. {λn−,λn+} are pairs of periodic and antiperiodic eigenvalues, and b∈(0,1) determines Gevrey classes.THEOREM.If∑n=1∞∣λn+−λn−∣2(1+n)2se2anb&amp;amp;amp;amp;amp;amp;lt;∞s⩾0, a0 , then V(x) is a Gevrey function, and moreover∑m=−∞∞∣v(m)∣2(1+∣m∣)2se2a∣m∣b&amp;amp;amp;amp;amp;amp;lt;∞.
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular pote... more We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L 2-potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p-spaces and other rearrangement invariant function spaces.
We consider the Hill operator Ly = −y ′′ + v(x)y, 0 ≤ x ≤ π, subject to periodic or antiperiodic ... more We consider the Hill operator Ly = −y ′′ + v(x)y, 0 ≤ x ≤ π, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the form v(x) = ae −2irx + be 2isx , a, b = 0, r, s ∈ N, r = s. It is shown that the system of root functions does not contain a basis in L 2 ([0, π], C) if bc are periodic or if bc are antiperiodic and r, s are odd or r = 1 and s ≥ 3.
Let us consider the Dirac operatorL=iJddx+U,J=100-1,U=0acos2πxacos2πx0,where a≠0 is real, on I=[0... more Let us consider the Dirac operatorL=iJddx+U,J=100-1,U=0acos2πxacos2πx0,where a≠0 is real, on I=[0,1] with boundary conditions bc=Per+, i.e., F(1)=F(0), and bc=Per-, i.e., F(1)=-F(0), F=f1f2∈H1(I). Then σ(Lbc)=-σ(Lbc), and all λ∈σPer+(L(U)) are of multiplicity 2, while λ∈σPer-(L(U)) are simple (Theorem 15). This is an analogue of Ince&amp;#39;s statement for Mathieu–Hill operator.Links between the spectra of Dirac and Hill operators lead to detailed information about
We give a complete description of the structure of the spectra of Hill operator Ly = −y + (a cos ... more We give a complete description of the structure of the spectra of Hill operator Ly = −y + (a cos 2x + b cos 4x)y, a, b real, x ∈ [0, ] with periodic or antiperiodic boundary conditions. As in Ince [Proc. London Math. Soc. 23 (1923) 56] and Magnus-Winkler [Hill's Equation, Interscience Publishers, Wiley, 1969], properties and spectra of special tridiagonal matrices is a core of our analysis.
It is proved using positive definite functions that a normed space X is unifomly homeomorphic to ... more It is proved using positive definite functions that a normed space X is unifomly homeomorphic to a subset of a Hilbert space, if and only if X is (linearly) isomorphic to a subspace of a L,,(/z) space (= the space of the measurable functions on a probability space with convergence in probability). As a result we get that lp (respectively L,(0,1)), 2<p <~, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces.
A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] l... more A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] led to an integral operator K-see (1), (2). They guessed that the equation E 0 (a, T) = 1 where E 0 is the largest eigenvalue of the operator K has a solution (*) T (a) = 1 − τ (a) with τ (a) ∼ a 2/5 when a goes to 0. τ (a) imitates the shift of critical (instability) temperature. We give a rigorous analysis of an anisotropic integral operator K and prove the asymptotic (*)-see Theorem 8 and Proposition 10. Additive Uncertainty Principle (of Landau-Pollack-Slepian [SP], [LP1, LP2]) plays important role in this analysis.
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular pote... more We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L 2-potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p-spaces and other rearrangement invariant function spaces.
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